Questions — OCR MEI C3 (366 questions)

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OCR MEI C3 2008 June Q5
5 Show that the curve \(y = x ^ { 2 } \ln x\) has a stationary point when \(x = \frac { 1 } { \sqrt { \mathrm { e } } }\).
OCR MEI C3 2008 June Q6
6 In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$
  1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
  2. Find the time when the mass is 30 grams.
  3. Sketch the graph of \(m\) against \(t\).
OCR MEI C3 2008 June Q7
7 Given that \(x ^ { 2 } + x y + y ^ { 2 } = 12\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR MEI C3 2008 June Q8
8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 1 + \cos x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
P is the point on the curve with \(x\)-coordinate \(\frac { 1 } { 3 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8feffafd-4eba-4968-b4d2-88fa364d6170-3_825_816_571_662} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(y\)-coordinate of P .
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the gradient of the curve at the point P .
  3. Show that the derivative of \(\frac { \sin x } { 1 + \cos x }\) is \(\frac { 1 } { 1 + \cos x }\). Hence find the exact area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 3 } \pi\).
  4. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arccos \left( \frac { 1 } { x } - 1 \right)\). State the domain of this inverse function, and add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
OCR MEI C3 2008 June Q9
9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt { 4 - x ^ { 2 } }\) for \(- 2 \leqslant x \leqslant 2\).
  1. Show that the curve \(y = \sqrt { 4 - x ^ { 2 } }\) is a semicircle of radius 2 , and explain why it is not the whole of this circle. Fig. 9 shows a point \(\mathrm { P } ( a , b )\) on the semicircle. The tangent at P is shown. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8feffafd-4eba-4968-b4d2-88fa364d6170-4_625_933_589_607} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  2. (A) Use the gradient of OP to find the gradient of the tangent at P in terms of \(a\) and \(b\).
    (B) Differentiate \(\sqrt { 4 - x ^ { 2 } }\) and deduce the value of \(\mathrm { f } ^ { \prime } ( a )\).
    (C) Show that your answers to parts ( \(A\) ) and ( \(B\) ) are equivalent. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } ( x - 2 )\), for \(0 \leqslant x \leqslant 4\).
  3. Describe a sequence of two transformations that would map the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\). Hence sketch the curve \(y = \mathrm { g } ( x )\).
  4. Show that if \(y = \mathrm { g } ( x )\) then \(9 x ^ { 2 } + y ^ { 2 } = 36 x\).
OCR MEI C3 2010 June Q1
1 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos 3 x \mathrm {~d} x\).
OCR MEI C3 2010 June Q2
2 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.
OCR MEI C3 2010 June Q3
3
  1. Differentiate \(\sqrt { 1 + 3 x ^ { 2 } }\).
  2. Hence show that the derivative of \(x \sqrt { 1 + 3 x ^ { 2 } }\) is \(\frac { 1 + 6 x ^ { 2 } } { \sqrt { 1 + 3 x ^ { 2 } } }\).
OCR MEI C3 2010 June Q4
4 A piston can slide inside a tube which is closed at one end and encloses a quantity of gas (see Fig. 4). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{30d0d728-d6d6-4a54-baf9-a6df8646bf64-2_154_1003_1080_571} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The pressure of the gas in atmospheric units is given by \(p = \frac { 100 } { x }\), where \(x \mathrm {~cm}\) is the distance of the piston from the closed end. At a certain moment, \(x = 50\), and the piston is being pulled away from the closed end at 10 cm per minute. At what rate is the pressure changing at that time?
OCR MEI C3 2010 June Q5
5 Given that \(y ^ { 3 } = x y - x ^ { 2 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y ^ { 2 } - x }\).
Hence show that the curve \(y ^ { 3 } = x y - x ^ { 2 }\) has a stationary point when \(x = \frac { 1 } { 8 }\).
OCR MEI C3 2010 June Q6
6 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 1 + 2 \sin 3 x , \quad - \frac { \pi } { 6 } \leqslant x \leqslant \frac { \pi } { 6 }$$ You are given that this function has an inverse, \(\mathrm { f } ^ { - 1 } ( x )\).
Find \(\mathrm { f } ^ { - 1 } ( x )\) and its domain.
OCR MEI C3 2010 June Q7
7 State whether the following statements are true or false; if false, provide a counter-example.
  1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
  2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
  3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational.
OCR MEI C3 2010 June Q8
8 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{30d0d728-d6d6-4a54-baf9-a6df8646bf64-3_730_841_561_651} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Verify that the coordinates of P are \(( 1,0 )\).
  2. Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
  3. Find \(\int \ln x \mathrm {~d} x\). Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.
OCR MEI C3 2010 June Q9
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). The curve crosses the \(y\)-axis at P. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{30d0d728-d6d6-4a54-baf9-a6df8646bf64-4_604_1233_358_456} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the coordinates of P .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer. Hence calculate the gradient of the curve at P .
  3. Show that the area of the region enclosed by \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is
    \(\frac { 1 } { 2 } \ln \left( \frac { 1 + \mathrm { e } ^ { 2 } } { 2 } \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { 2 } \left( \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } \right)\).
  4. Prove algebraically that \(\mathrm { g } ( x )\) is an odd function. Interpret this result graphically.
  5. (A) Show that \(\mathrm { g } ( x ) + \frac { 1 } { 2 } = \mathrm { f } ( x )\).
    (B) Describe the transformation which maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { f } ( x )\).
    (C) What can you conclude about the symmetry of the curve \(y = \mathrm { f } ( x )\) ?
OCR MEI C3 Q1
1 Prove that the product of consecutive integers is always even.
OCR MEI C3 Q2
2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { 1 + x ^ { 3 } }\).
OCR MEI C3 Q3
3 The graph shows part of the function \(y = a \ln ( b x )\).
\includegraphics[max width=\textwidth, alt={}, center]{2f403099-2813-40d8-a9ae-1f7e64d41f80-2_377_762_900_685} The graph passes through the points \(( 2,0 )\) and \(( 4,1 )\).
  1. Show that \(b = \frac { 1 } { 2 }\) and find the exact value of \(a\).
  2. Solve the inequality \(| a \ln ( b x ) | < 2\).
OCR MEI C3 Q4
4
  1. Show that \(y = a x e ^ { - x }\) for \(a > 0\) has only one stationary point for all values of \(x\). Determine whether this stationary value is a maximum or minimum point.
  2. Sketch the curve.
OCR MEI C3 Q5
5 Find \(\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\), giving your answer to 1 decimal place.
OCR MEI C3 Q6
6 Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x )\) and hence or otherwise find the value of \(\int _ { 2 } ^ { 3 } \ln x \mathrm {~d} x\), giving your answer in the form \(\ln a + b\), where \(a\) and \(b\) are to be determined.
OCR MEI C3 Q7
7 Two quantities, \(x\) and \(\theta\), vary with time and are related by the equation \(x = 5 \sin \theta - 4 \cos \theta\).
  1. Find the value of \(x\) when \(\theta = \frac { \pi } { 2 }\).
  2. When \(\theta = \frac { \pi } { 2 }\), its rate of increase (in suitable units) is given by \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = 0.1\). Show that at that moment \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4\).
OCR MEI C3 Q8
8 You are given that \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 1 }\) for all real values of \(x\).
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\).
  2. Hence show that there is a stationary value at \(\left( 1 , \frac { 1 } { 2 } \right)\) and find the coordinates of the other stationary point.
  3. The graph of the curve is shown in Fig. 8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-3_518_892_1612_705} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure} State whether the curve is odd or even and prove the result algebraically.
  4. Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x = \int _ { a } ^ { b } k \frac { 1 } { u + 1 } \mathrm {~d} u\), where the values of \(a , b\) and \(k\) are to be determined.
  5. Hence find the area of the shaded region in Fig. 8.
OCR MEI C3 Q9
9 The curve in Fig. 9.1 has equation \(\sqrt { x } + \sqrt { y } = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_426_647_299_667} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Show that this is part, but not all of the curve \(y = 1 - 2 \sqrt { x } + x\). Sketch the full curve \(y = 1 - 2 \sqrt { x } + x\).
  2. Fig.9.2 shows a star shape made up of four parts, one of which is given in part (i) above. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_380_681_1197_651} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} For each of the sections of the shape labelled \(\mathrm { A } , \mathrm { B }\) and C , state the equation of the curve and the domain.
  3. The shape shown in Fig.9.2 is made into that in Fig. 10.3 by stretching the part of the figure for which \(y > 0\) by a scale factor of 2 . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_405_686_1996_605} \captionsetup{labelformat=empty} \caption{Fig. 9.3}
    \end{figure} Find the area of this shape.
OCR MEI C3 Q1
1 Prove that the product of any three consecutive integers is a multiple of 6 .
OCR MEI C3 Q2
2
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Hence, or otherwise, solve the inequality \(| 2 x - 3 | < 5\). Illustrate your answer on your graph.